Curved Spacetime

General relativity replaces the fixed Minkowski metric $\eta_{\mu\nu}$ with a position-dependent Lorentzian metric $g_{\mu\nu}(x)$ on a differentiable manifold. Gravity is not a force in the Newtonian sense — it is the geometry of spacetime, encoded in $g_{\mu\nu}$ and its derivatives.

From flat to curved

In special relativity, free particles follow straight worldlines in Minkowski space. In GR, free particles follow geodesics of $(\mathcal{M}, g)$:

$$ \frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = 0 $$

The Christoffel symbols (connection coefficients) are

$$ \Gamma^\mu_{\alpha\beta} = \frac{1}{2} g^{\mu\nu}\left(\partial_\alpha g_{\nu\beta} + \partial_\beta g_{\nu\alpha} - \partial_\nu g_{\alpha\beta}\right) $$

In flat spacetime with inertial coordinates, $g_{\mu\nu} = \eta_{\mu\nu}$ and $\Gamma^\mu_{\alpha\beta} = 0$ — geodesics reduce to straight lines.

The Einstein field equations

Matter and energy curve spacetime via

$$ G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G\thinspace T_{\mu\nu} $$

where

$$ G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} R\thinspace g_{\mu\nu} $$

is the Einstein tensor, $R_{\mu\nu}$ the Ricci tensor, $R$ the Ricci scalar, $\Lambda$ the cosmological constant, and $T_{\mu\nu}$ the stress-energy tensor.

In vacuum ($T_{\mu\nu} = 0$), the field equations become $R_{\mu\nu} = 0$ (when $\Lambda = 0$) — spacetime can still be curved (Schwarzschild solution).

Riemann curvature

The Riemann tensor measures intrinsic curvature:

$$ R^\rho_{\ \sigma\mu\nu} = \partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma} $$

Key symmetries: $R_{\rho\sigma\mu\nu} = -R_{\sigma\rho\mu\nu}$, $R_{\rho\sigma\mu\nu} = R_{\mu\nu\rho\sigma}$, and the Bianchi identity $R_{\rho[\sigma\mu\nu]} = 0$.

Flat spacetime has $R^\rho_{\ \sigma\mu\nu} = 0$ globally; locally, any smooth Lorentzian manifold is approximately Minkowski in a small enough neighborhood (equivalence principle).

Relation to Minkowski space

Every Lorentzian manifold has, at each event $p$, a tangent space $T_p\mathcal{M}$ equipped with Minkowski inner product $g_{\mu\nu}(p)$. Special relativity governs physics in that infinitesimal neighborhood; global curvature accumulates over finite distances.

Schwarzschild metric (spherical symmetry)

Outside a spherically symmetric mass $M$ (in geometric units $G = c = 1$):

$$ ds^2 = -\left(1 - \frac{2M}{r}\right) dt^2 + \left(1 - \frac{2M}{r}\right)^{-1} dr^2 + r^2 d\Omega^2 $$

where $d\Omega^2 = d\theta^2 + \sin^2\theta\thinspace d\phi^2$.

• Event horizon at $r = 2M$
• Gravitational redshift — clocks run slower in stronger gravitational fields
• Perihelion precession — Mercury’s orbit precesses because $R \neq 0$

Geodesic deviation (tidal forces)

Nearby geodesics do not remain parallel in curved spacetime. The separation vector $V^\mu$ between two geodesics satisfies

$$ \frac{D^2 V^\mu}{d\tau^2} = -R^\mu_{\ \nu\rho\sigma} u^\nu V^\rho u^\sigma $$

This is the relativistic origin of tidal gravity — the Riemann tensor directly measures how a gravitational field differs from place to place.

Curvature and physics pipeline

flowchart TD
    T[Tμν stress-energy] --> EFE[Einstein field equations]
    EFE --> G[gμν metric]
    G --> CHR[Christoffel symbols Γ]
    CHR --> GEO[Geodesic equation]
    G --> RIE[Riemann tensor]
    RIE --> TID[Tidal forces / deviation]
    GEO --> OBS[Observable orbits, redshift, lensing]
    TID --> OBS

Weak-field limit

For weak, static fields ($|\Phi| \ll 1$, slow motion), write

$$ g_{00} \approx -(1 + 2\Phi), \quad g_{ij} \approx (1 - 2\Phi)\delta_{ij} $$

where $\Phi$ is the Newtonian gravitational potential. The geodesic equation reproduces $\ddot{\mathbf{x}} = -\nabla\Phi$ — Newtonian gravity emerges as a limit.

Why curved spaces appear in physics

1. Equivalence principle — gravity is locally indistinguishable from acceleration; globally it requires curved geometry.
2. Coordinate independence — tensor equations on manifolds express laws valid in any smooth coordinate chart.
3. Cosmology — the universe’s large-scale structure is modeled as a time-dependent curved (often Friedmann–Lemaître–Robertson–Walker) spacetime.
4. Black holes & waves — regions where $g_{\mu\nu}$ deviates radically from $\eta_{\mu\nu}$ are observational targets (LIGO, EHT).

Minkowski space is the linearized, zero-curvature backbone; curved Lorentzian geometry is the dynamical stage on which matter writes its gravitational history.